In the computer graphics (CG) field, there are various methods for obtaining correspondences between two shape models represented by triangular mesh elements in morphing or the like. There are also a method for obtaining correspondences between points among all points and a method for obtaining correspondences between all the points.
Surface parameterization is a method for mapping (or executing bijection) a three-dimensionally spatial surface to a certain region (plain surface, sphere, or the like) and is used in order to execute texture mapping or re-meshing in the CG field. The surface parameterization is generally executed in order to map such a surface to a plain space in many cases and map such a surface to a spherical space in some cases.
For example, it is assumed that the surface parameterization is executed so as to map two shapes (surfaces) represented by triangular mesh elements to a spherical space, calculate corresponding points on the same spherical space, and execute mapping between the two shapes.
In this case, it is assumed that the two shapes have label information items in regions that are defined for the shapes, respectively. In order to appropriately map the label information items, boundaries between regions for which different label information items are provided overlap (or match) each other on a spherical surface to which the shapes have been mapped. However, the boundaries may not be appropriately mapped to the spherical surface by the conventional surface parameterization.
A case where a hand model A illustrated in FIG. 1 is mapped to a hand model B illustrated in FIG. 2 is considered. Finger labels are added to fingers of the hand models A and B, while palm labels are added to palms of the hand models A and B. A spherical model A is generated by mapping the hand model A to a spherical surface illustrated in FIG. 3, while a spherical model B is generated by mapping the hand model B to a spherical surface illustrated in FIG. 4. After that, the spherical model A is mapped to the spherical model B so as to ensure that boundaries of regions to which the finger labels are added in the spherical models A and B match each other.
The conventional surface parameterization supports mapping of a surface to a spherical surface. In this case, the position of a node xiεX is calculated by minimizing the following energy based on a spring model having a spring constant wij between the node xi and a node xj.
                              X          =                      arg            ⁢                                                  ⁢                                          min                X                            ⁢                                                ∑                  i                                ⁢                                                                  ⁢                                                      ∑                                          j                      ∈                                              N                        ⁡                                                  (                          i                          )                                                                                                      ⁢                                                                          ⁢                                                            w                      ij                                        ⁢                                                                                                                                                x                            i                                                    -                                                      x                            j                                                                                                                      2                                                                                                          ⁢                                  ⁢                              s            .            t            .                                                  ⁢                                                        x                i                                                            =          1                                    (        1        )            
Since the spherical surface is a surface, a method that is the same as or similar to the aforementioned method may be used for mapping of a spherical surface to another spherical surface. However, the position of the node xiεX is calculated by minimizing, based on the spring model, the energy that causes nodes to be placed on the spherical surface under a constraint that boundaries between regions of the spherical model A are fixed to boundaries between corresponding regions of the spherical model B.
Although it looks as if approximately appropriate process results are obtained as illustrated in FIG. 5, microscopic white parts are generated at a boundary between the thumb and the palm as illustrated in FIG. 6, and microscopic white parts are generated at a boundary between the little finger and the palm as illustrated in FIG. 7. The white parts generated at the boundaries indicate that triangular mesh elements are reversed.
FIG. 8 illustrates normal triangular elements of a mesh, while FIG. 9 illustrates the state of a reversed part in which a node 1000 is placed on a triangular element 200.
Since the energy according to the conventional technique is based on the spring model, the energy may be stable in a state in which a spring is reversed. In the state illustrated in FIG. 9, however, bijection is not achieved and a label may not be mapped.
Examples of related art are “XIANFENG GU and SHING-TUNG YAU, “COMPUTING CONFORMAL STRUCTURES OF SURFACES”, COMMUNICATIONS IN INFORMATION AND SYSTEMS, Vol. 2, pp. 121-146, December, 2002” and “Shadi Saba, Irad Yavneh, Craig Gotsman and Alla Sheffer, “Practical Spherical Embedding of Manifold Triangle Meshes”, Proceedings of the International Conference on Shape Modeling and Applications (SMI'05), 2005”.
According to an aspect, an object of the disclosure is to provide a technique for mapping a first model to a second model while achieving bijection.